How do you read this logical statement and verify its truth value? Nested quantifiers

Question

$\forall x\exists y \forall z ((y>0) \land ((z^2<y) \rightarrow (z^2+1<x^4)))$

And also how would you verify quantifier claims over a domain, ie. reals? I previously have been doing these problems as just running over some possible cases, with no set technique. Is that the right way to do these problems?

Answer 1

For every $x$ there exists a $y>0$ such that for every $z$ if $z^2<y$ then $z^2+1<x^4$

Answer 2

For all $x$, there exist $y$, such that for all $z$, if $y$ is greater than $0$ and $z$ square is less than $y$, then $z$ square plus $1$ is less than $x$ to the four power.

Answer 3

For all $x$, there is a $y$ that is positive and for every $z$, if $z^2<y$ then $z^2+1<x^4$.

Answer 4

You have: $ \forall x \exists y \forall z\,(y>0 \wedge (z^2< y \to z^2+1<x^4) \\ \forall x \exists y \forall z\,(y>0 \wedge \neg(z^2< y \wedge z^2+1\geq x^4) \\ \forall x \exists y (y>0 \wedge \neg\exists z\,(x^4-1\leq z^2< y) $

For any $x$ there is a positive $y$ such that there is no $z$ such that $z^2$ is on the interval $[x^4-1; y)$.

Is this true?

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PS: What domain are you discussing? The real numbers? rationals? integers?